Regularization-Based Constrained Distributed Bilevel Optimization Over Unbalanced Graphs

This paper focuses on solving a class of constrained distributed bilevel optimization problems over unbalanced graphs, where all agents are equipped with convex inner objective functions and strongly convex outer ones. The goal of solving the considered bilevel optimization problem is to minimize the global objective functions at both levels. In this paper, we propose a regularization-based distributed projected algorithm with row stochastic matrices and a time-varying regularization parameter $\theta _{t}$ . Furthermore, with the aid of the strong convexity of outer objective functions and the smoothness of two level objective functions, we establish that the proposed algorithm converges to the optimal solution with ${\mathcal {O}}(t^{-a+b})$ ( $a\in (0.5,1)$ and $b\in (0,0.5)$ ) and ${\mathcal {O}}(t^{-b})$ convergence rates from the perspectives of the outer and inner objective functions, respectively. Finally, we illustrate the effectiveness of the proposed algorithm by numerical simulations. Note to Practitioners—Constrained distributed bilevel optimization has garnered increasing attention in automation control and scientific computing, thanks to its wide range of applications in microgrids, wireless sensor networks, artificial intelligence, and other emerging industries. In real-world scenarios, bilevel optimization offers a broader modeling scope than general single-level optimization problems. Bilevel formulations can capture a variety of existing optimization challenges, including constrained nonlinear and ill-posed constrained optimization. However, this generality also introduces substantial analytical complexities. To address these issues, this paper presents an efficient distributed algorithm that leverages a regularization method and a mild assumption on the networked communication topology (specifically, a row-stochastic weighted matrix) for constrained bilevel optimization. Unlike conventional centralized algorithms and their distributed counterparts, which often rely on restrictive assumptions (such as balanced graphs on networked topologies), the proposed algorithm requires only weak conditions to enable each agent to cooperatively achieve the global optimum. Meanwhile, we establish convergence results for the decision variables and convergence rates with respect to the outer and inner objective functions. The validity and correctness of the proposed method are demonstrated through a sensor network problem and a real-world image recovery problem. Our future work will focus on designing distributed algorithms with improved convergence performance using accelerated techniques.